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Boatman's Conjecture.
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Boatman's Conjecture
This simplified reasoning or blog is based on
exact initial condition and results are definite:
The boat will not return to initial place
no matter how you jump!
B x V1 (supposed equivalently lost)
= B x ΔV
= (M + BM) x V2 (=> V1> V2)
where B is the boat mass,
BM is the mass of boatman.
B and BM can not reach the original place
because there is a place mismatch.
For this problem, the resistance can
have any type of velocity function,
the result is the same.
However, the boat nearly never return to the origin!?
It's not that after 'infinite time', it will reach the origin,
far from it, there is a 'long distance' from the origin
that you can not ignore.
When initial condition is given,
everything can be determined exactly.
Notice the following:
1. Immediately after jump, the momentum
conservation theorem can be used, minus k Xj,
where Xj is the distance for the boat getting
its initial momentum, ideally reaching zero.
2. Immediately after landing, the momentum
conservation theorem can still be used, minus k Xl,
where Xl is the distance for the boat changing
its momentum, ideally reaching zero. Even though
Xj and Xl are big enough, momentum conservation
theorem is always true for this problem.
3. The place of jump play the important role of
final boat position. The place of resistance and
the places of objects are different. As you will
choose a place for calculating its initial momentum,
otherwise you will not get even a demonstration
momentum, you will also choose a place for the
final momentum, and by the theorem, you can only choose
one place. By the law of the nature, this is also true.
4. Static resistance and other influences
can be considered, this does not prevent the use of
conservation theorem, with the result slightly changed.
Strictly obeying the laws of physics is the answer.
This is a very special example.
Finally, a more generic conjecture can be proposed
Boatman's Conjecture
A boatman jump forward, for any rational
resistance, after stabalizaton, the boat will
not return to its initial place.
Good luck.
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